In this paper we take an idea presented in recent paper by Carlen, Carvalho, Le Roux, Loss, and Villani ([3]) and push it one step forward to find an exact estimation on the entropy production. The new estimation essentially ...

The sharp trace inequality of José Escobar is extended to traces for the fractional Laplacian on Rⁿ and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb’s ...

Cayley submanifolds of R^8 were introduced by Harvey and Lawson as an instance of calibrated submanifolds, extending the volume-minimising properties of complex submanifolds in Kähler manifolds. More generally, Cayley ...

The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers Symn of a cuspidal representation of GL.2/ over the adèles of F , where F is a number field. In 1978, Gelbart and ...

We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for GL_2.

We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampere equation. We show that for certain boundary data on P^1 the solution Φ to this Dirichlet problem is connected ...

This thesis studies the structure of categories of polynomials, the diagrams that represent polynomial functors. Specifically, we construct new models of intensional dependent type theory based on these categories. Firstly, ...

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds ...

We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his ...

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if G is a virtually compact special hyperbolic group, and Q ≤ G is a K-quasiconvex ...

Let V be a fi nite graph and let ∅ : V → V be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group G. Then G acts freely and cocompactly on a CAT(0) cube complex. Hence, if F is a ...

The current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was ...

Motivated by the problem of inferring the molecular dynamics of DNA in solution, and linking them with its base-pair composition, we consider the problem of comparing the dynamics of functional time series (FTS), and of ...

The Nielsen–Schreier theorem states that subgroups of free groups are free. As all of its proofs use the Axiom of Choice, it is natural to ask whether the theorem is equivalent to the Axiom of Choice. Other questions arise ...

For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of ...

We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill. The proof involves passing to an ...

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many ...

We consider an h-partite version of Dilworth's theorem with multiple partial orders. Let P be a fi nite set, and let <₁, ..., <ᵣ be partial orders on P. Let G(P, <₁, ..., <ᵣ) be the graph whose vertices are the elements ...

In this dissertation, we treat several problems in Ramsey theory, probabilistic combinatorics and extremal graph theory.